ANCOVA allows you to remove from a dependent variable (y) irrelevant or error variance that can not be predicted from your independent variable (x). Hence, by accounting for the third variable, you are more able to obtain a accurate picture of the proportion of variance in y that x is capable of accounting for; in other words, your power is increased.
Two general applications exist for ANCOVA:
Remove Error Variance in the Randomized Experiment: Participants are assigned to treatment and control groups in any ANOVA-type design. ANCOVA is then used as the statistical technique to eliminate irrelevant y variance.
Equating Non-Equivalent (Intact) Groups: A very controversial use of ANCOVA is to correct for initial group differences (prior to assigned to x) that exists on y among several intact, state variable groups.
Assumptions of ANCOVA
Traditional ANCOVA, like other methods, relies on several assumptions:
The Covariate: When you reference back to the previous page on extraneous variables, you will note that the definition of a covariate assumes that the third variable (z) is unrelated to x, and that z is related to, and in a sense, acts as a suppressor of y.
Linearity: Since ANCOVA is a general linear model procedure with much in common with multiple regression, it is also assumed that the covariate has a linear relationship with the dependent variable.
Homogeneity of Variance: Like previous techniques, ANCOVA assumes homogeneity of variance. In other words, the variance of group one is equal to the variance of group 2 and so on.
Homogeneity of Regression: ANCOVA assumes that homogeneity of regression exists--that the correlation between y and z is equal for all levels of x. In other words, for each level of the independent variable, the slope of the prediction of the dependent variable from the covariate must be equal.
http://www.uwsp.edu/psych/cw/statmanual/ancovaoverview.html
Two general applications exist for ANCOVA: